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Easy-plane anisotropy makes spin orientations within one plane energetically preferable to orientations normal to it. For example, a classical single-site term $D(S^z{)}^2$ with $D>0$ penalizes out-of-plane components and leaves the $xy$ plane degenerate. At sufficiently low energies, an in-plane angle can then be the main orientational degree of freedom.

The resulting approximate angular description connects a magnet to the XY model, but the microscopic form matters. For a quantum spin one-half, $(S^z{)}^2$ is proportional to the identity, so that single-site term alone does not generate orientational anisotropy. Anisotropic exchange interactions provide another route. Stating the spin size and Quantum Hamiltonian prevents transferring a classical mechanism to a quantum model where it is ineffective. Residual in-plane anisotropy can further select a finite set of directions rather than a continuous circle.

Write $\theta =\pi /2-\vartheta$ and $\varphi =\phi$. To quadratic order,After dropping the total derivative $S{\phi }_t$, the quadratic Lagrangian density isIts Euler--Lagrange equations areFor modes proportional to $e^{i(kx-\omega t)}$, nontrivial amplitudes requireThusAt short wavelength the anisotropy is negligible and the quadratic dispersion is that of a conventional ferromagnetic magnon. At long wavelength, easy-plane anisotropy makes the out-of-plane fluctuation the conjugate density of the in-plane phase; eliminating it produces the linear Goldstone sound mode characteristic of a superfluid.